Let $x , y$ and $z$ be positive real numbers. Suppose $x , y$ and $z$ are the lengths of the sides of a triangle opposite to its angles $X , Y$ and $Z$, respectively. If
$$\tan \frac { X } { 2 } + \tan \frac { Z } { 2 } = \frac { 2 y } { x + y + z }$$
then which of the following statements is/are TRUE?
(A) $2 Y = X + Z$
(B) $Y = X + Z$
(C) $\tan \frac { X } { 2 } = \frac { x } { y + z }$
(D) $x ^ { 2 } + z ^ { 2 } - y ^ { 2 } = x z$

In a $\triangle PQR$, if $3\sin P + 4\cos Q = 6$ and $4\sin Q + 3\cos P = 1$, then the angle $R$ is equal to
(1) $\frac{5\pi}{6}$
(2) $\frac{\pi}{6}$
(3) $\frac{\pi}{4}$
(4) $\frac{3\pi}{4}$

In a triangle $A B C$, if $\cos A + 2 \cos B + \cos C = 2$ and the lengths of the sides opposite to the angles $A$ and $C$ are 3 and 7 respectively, then $\cos A - \cos C$ is equal to
(1) $\frac { 9 } { 7 }$
(2) $\frac { 10 } { 7 }$
(3) $\frac { 5 } { 7 }$
(4) $\frac { 3 } { 7 }$

$ABC$ triangle, AFD equilateral triangle, $[DE]$ // $[AB]$, $m ( \widehat { DFC } ) = x$
In the figure, $m \widehat { ( \mathrm { ACF } ) } = m \widehat { ( \mathrm { FCB } ) } = m \widehat { ( \mathrm { DEC } ) }$ and points $D$, $E$, $F$ lie on the sides of triangle ABC.
Accordingly, what is x in degrees?\ A) 20\ B) 25\ C) 30\ D) 35\ E) 40

ABC is a triangle $$\begin{aligned}& | \mathrm { AD } | = | \mathrm { CD } | = | \mathrm { BC } | \\& \mathrm { m } ( \widehat { \mathrm { BAD } } ) = 20 ^ { \circ } \\& \mathrm { m } ( \widehat { \mathrm { BCD } } ) = 60 ^ { \circ } \\& \mathrm { m } ( \widehat { \mathrm { ACD } } ) = \mathrm { x }\end{aligned}$$ Accordingly, what is x in degrees?\ A) 5\ B) 10\ C) 15\ D) 20\ E) 25

In the figure, a regular hexagon and a square sharing one side are given. A regular polygon sharing one side with the regular hexagon and one side with the square is to be drawn as shown in the figure.
Accordingly, how many sides does the regular polygon to be drawn have?
A) 10
B) 12
C) 15
D) 16
E) 18

A triangular ABC cardboard with vertices labeled with letters A, B, and C is shown as in Figure 1. 3 ABC cardboards can be assembled on a flat surface as shown in Figure 2 by overlapping the A vertices and leaving no gaps between the edges and without the cardboards overlapping.
The same process can be done using 9 ABC cardboards by overlapping the B vertices.
Accordingly, using how many ABC cardboards can this process be done by overlapping the C vertices?
A) 10
B) 12
C) 15
D) 18
E) 20

One interior angle of an $n$-sided regular polygon is calculated as $\frac{(n-2) \cdot 180^{\circ}}{n}$.
A triangular piece of paper is cut along the dashed lines as shown in the figure, 3 triangular pieces are removed, and a regular hexagon is obtained.
Given that the sum of the perimeters of the removed triangles is 36 units, what is the perimeter of the hexagon?
A) 18
B) 24
C) 30
D) 36
E) 42